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				   explain differential equation behind wave motion.


6 years ago

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										Mathematically, the most basic wave is the sine  wave (or harmonic wave or sinusoid), with an amplitude u described by the equation:
$u(x, \ t)= A \cos (kx - \omega t + \phi) \ ,$
where A is the semi-amplitude of the wave, half the peak-to-peak  amplitude, often called simply the amplitude – the maximum distance from  the highest point of the disturbance in the medium (the crest) to the  equilibrium point during one wave cycle. In the illustration to the  right, this is the maximum vertical distance between the baseline and  the wave; x is the space coordinate, t is the time  coordinate, k is the wavenumber  (spatial frequency), ω is the temporal frequency, and φ is a phase offset.
The wave equation
The wave equation is a partial differential equation  that describes the evolution of a wave over time in a medium where the  wave propagates at the same speed independent of wavelength , and independent of  amplitude .
consider the wave equation in one dimension, for  example, as applied to a string. Suppose a one-dimensional wave is  traveling along the x axis with velocity v and amplitude u (which generally  depends on both x and t), the wave equation is
$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,$
The velocity v will depend on the medium through which the  wave is moving.
The general solution for the wave equation
$u(x,t)=F(x-vt)+G(x+vt). \,$
This formula represents two shapes traveling through the medium in  opposite directions; F in the positive x direction, and G in the negative x direction, of arbitrary functional shapes F and G.
--
regards
Ramesh

6 years ago

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